MAHLER’S MEASURE AND SPECIAL VALUES OF L-FUNCTIONS

DAVID W. BOYD

Abstract. If P (x_{1}, …, x_{n}) is a polynomial with integer coefficients, the Mahler measure M(P) of P is defined to be the geometric mean of |P | over the n-torus 𝕋^{n}. For n = 1, M(P) is an algebraic integer, but for n > 1, there is reason to believe that M(P) is usually transcendental. For example, Smyth showed that log M(1 + x + y) = L′(χ_{−3}, −1), where χ_{−3} is the odd Dirichlet character of conductor 3. Here we will describe some examples for which it appears that log M(P (x, y)) = r L′(E, 0), where E is an elliptic curve and r is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.